A unified high-order model for size-dependent vibration of nanobeam based on nonlocal strain/stress gradient elasticity with surface effect

In this work, a unified high-order nanobeam model considering various high-order shear deformation beam theories is established to investigate the vibration response of nanobeam on the basis of two-phase local/nonlocal strain and stress gradient theory, as well as surface elasticity theory. The unified model also includes the Euler-Bernoulli beam model and Timoshenko beam model. Herein, the elastic dynamics governing equations and boundary conditions are derived using Hamilton's principle, and the analytical solutions, such as exact formulas for natural frequencies, are obtained by employing the Navier method for simply supported boundary conditions. The effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation and surface energy in stress and strain-driven models are analyzed in detail, respectively. The parametric studies reveal that the two scale parameters (nonlocal parameters and material length characteristic parameters) have opposite effects on the stiffness of the nanobeams in the two driving models, while the surface parameters have the same effect on the stiffness of the two driving models. The influence of the slenderness ratio on the surface effect and scale effect is opposite, meaning that the increase of the slenderness ratio deepens the influence of the surface effect but weakens the influence of the scale effect. There are also differences in the effects of higher-order modes on the two effects. Higher modes lead to more significant scale effects, but the effect of higher modes on surface effects depends on the surface elastic properties of the material. We also find that the introduction of surface elasticity increases the gap between the TBT and other higher-order beams, which indicates the prediction results of the higher-order beam model are more accurate when both surface and nonlocal effects are considered. In addition, it is represented that the surface elasticity makes aluminum nanobeams exhibit a stiffness softening effect, while the effect of surface elasticity on the stiffness of silicon nanobeams is significantly dependent on the slenderness ratio and the number of modes.